Nagata's conjecture on curves
In mathematics, the Nagata conjecture on curves, named after Masayoshi Nagata, governs the minimal degree required for a plane algebraic curve to pass through a collection of very general points with prescribed multiplicities.
History
Nagata arrived at the conjecture via work on the 14th problem of Hilbert, which asks whether the invariant ring of a linear group action on the polynomial ring k[x1, ..., xn] over some field k is finitely generated. Nagata published the conjecture in a 1959 paper in the American Journal of Mathematics, in which he presented a counterexample to Hilbert's 14th problem.
Statement
- Nagata Conjecture. Suppose p1, ..., pr are very general points in P2 and that m1, ..., mr are given positive integers. Then for r > 9 any curve C in P2 that passes through each of the points pi with multiplicity mi must satisfy
- [math]\displaystyle{ \deg C \gt \frac{1}{\sqrt{r}}\sum_{i=1}^r m_i. }[/math]
The condition r > 9 is necessary: The cases r > 9 and r ≤ 9 are distinguished by whether or not the anti-canonical bundle on the blowup of P2 at a collection of r points is nef. In the case where r ≤ 9, the cone theorem essentially gives a complete description of the cone of curves of the blow-up of the plane.
Current status
The only case when this is known to hold is when r is a perfect square, which was proved by Nagata. Despite much interest, the other cases remain open. A more modern formulation of this conjecture is often given in terms of Seshadri constants and has been generalised to other surfaces under the name of the Nagata–Biran conjecture.
References
- "On Nagata's conjecture", Journal of Algebra 236 (2): 692–702, 2001, doi:10.1006/jabr.2000.8515.
- Nagata, Masayoshi (1959), "On the 14-th problem of Hilbert", American Journal of Mathematics 81 (3): 766–772, doi:10.2307/2372927.
- Strycharz-Szemberg, Beata (2004), "Remarks on the Nagata conjecture", Serdica Mathematical Journal 30 (2–3): 405–430.
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